MathDB

1

Part of 2000 All-Russian Olympiad

Problems(3)

Quadratics with Common Roots

Source: All-Russian MO 2000

12/30/2012
Let a,b,ca,b,c be distinct numbers such that the equations x2+ax+1=0x^2+ax+1=0 and x2+bx+c=0x^2+bx+c=0 have a common real root, and the equations x2+x+a=0x^2+x+a=0 and x2+cx+bx^2+cx+b also have a common real root. Compute the sum a+b+ca+b+c.
quadraticsalgebra unsolvedalgebra
Sum floor 2^k/3 from k=0 to 100

Source: All-Russian MO 2000

12/30/2012
Evaluate the sum ⌊203⌋+⌊213⌋+⌊223⌋+⋯+⌊210003⌋. \left\lfloor \frac{2^0}{3} \right\rfloor + \left\lfloor \frac{2^1}{3} \right\rfloor + \left\lfloor \frac{2^2}{3} \right\rfloor + \cdots + \left\lfloor \frac{2^{1000}}{3} \right\rfloor.
floor functionalgebra unsolvedalgebra
Functions

Source: 0

2/21/2009
Find all functions f:R⟶R f: \mathbb{R}\longrightarrow \mathbb{R} such that f(x\plus{}y)\plus{}f(y\plus{}z)\plus{}f(z\plus{}x)\ge 3f(x\plus{}2y\plus{}3z) for all x,y,z∈Rx, y, z \in \mathbb R.
function